1、._ N 62 54913,. C,.O ;v“ ,NATIONAL ADVISORY C_ITTEE _FOR AERONAUTICSTECHNICAL NOTE 2913ON THE DEVELOPMENT OF TURBULENT WAKESFROM VORTEX STREETSBy Anatol RoshkoCalifornia Institute of TechnologyWashingtonMarch 1953Provided by IHSNot for ResaleNo reproduction or networking permitted without license fr
2、om IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-In NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSTECHNICAL NOTE 2913ON THE DEVELOPMENT OF TURBULENT WAKESFROM VORTEX STREETSBy Anatol RoshkoSUMMARYWake development behind circular cylinders at R
3、eynolds numbers from40 to I0,000 was investigated in a low-speed wind tunnel. Standard hot-wire techniques were used to study the velocity fluctuations.The Reynolds number range of periodic vortex shedding is dividedinto two distinct subranges. At R = 40 to IS0, called the stable range,regular vorte
4、x streets are formed and no turbulent motion is developed.The range R = iS0 to 300 is a transition range to a regime called theirregular range, in which turbulent velocity fluctuations accompany theperiodic formation of vortices. The turbulence is initiated by laminar-turbulent transition in the fre
5、e layers which spring from the separationpoints on the cylinder. This transition first occurs in the rangeR = 150 to 300.Spectrum and statistical measurements were made to study the velocityfluctuations. In the stable range the vortices decay by viscous diffusion.In the irregular range the diffusion
6、 is turbulent and the wake becomesfully turbulent in 40 to 50 diameters downstream.It was found that in the stable range the vortex street has a periodicspanwise structure.The dependence of shedding frequency on velocity was successfullyused to measure flow velocity.Measurements in the wake of a rin
7、g showed that an annular vortexstreet is developed.INTRODUCTIONIt is always difficult to determine precisely the date and authorof a discovery or idea. This seems to be the case with the periodicphenomena associated with flow about a cylinder. Although the effectProvided by IHSNot for ResaleNo repro
8、duction or networking permitted without license from IHS-,-,-2 NACATN 2913of wind in producin_ vibrations in wires (aeolian tones) had been knownfor sometime, the first experimental observations are due to Strouhal(reference i) who showedthat the frequency depends on the relative airvelocity and not
9、 the elastic properties of the wires. Soonafter,Rayleigh (1879, references 2 and 3) performed similar experiments. Hisformulation of the Reynolds numberdependencedemonstrates his remarkableinsight into the problem.These earliest observations were concerned with the relationsbetween vibration frequen
10、cy and wind velocity. The periodic nature ofthe wake was discovered later, although Leonardo da Vinci in the fifteenthcentury had already drawn somerather accurate sketches of the vortexformation in the flow behind bluff bodies (reference 4). However,Leonardos drawings show a symmetric row of vortic
11、es in the wake. Thefirst modernpictures showing the alternating arrangement of vorticesin the wake were published by Ahlborn in 1902 (reference 5); his visual-ization techniques have been used extensively since then. The importanceof this phenomenon, now known as the K_rm_ vortex street, was pointed
12、out by Benard (1908, reference 6).f In 1911 Karman gave his famous theory of the vortex street (refer-ence 7), stimulating a widespread and lasting series of investigationsof the subject. For the most part these concerned themselves withl experimental comparisons of real vortex streets with KArmans
13、idealizedmodel, calculations on the effects of various disturbances and configura-tions, and so on. It can hardly be said that any fundamental advance inthe problem has been made since K_rm_s stability papers, in which healso clearly outlined the nature of the phenomenon and the unsolvedproblems. Ou
14、tstanding perhaps is the problem of the periodic vortex-shedding mechanism, for which there is yet no suitable theoreticaltreatment.However, the results of the many vortex-street studies, especiallythe experimental ones, are very useful for further progress in the prob-lem. Attention should be drawn
15、 to the work of Fage and his associates(1927, references 8 to i0), whose experimental investigations were con-ducted at Reynolds numbers well above the ranges examined by most otherinvestigators. Their measurements in the wake close behind a cylinderprovide much useful information about the nature o
16、f the shedding. Morerecently Kovasznay (1949, reference ii) has conducted a hot-wire inves-tigation of the stable vortex street (low Reynolds numbers), to whichfrequent reference will be made.Vortex-street patterns which are stable and well-defined for longdistances downstream actually occur in only
17、 a small range of cylinderReynolds numbers, from about R = 40 to 150, and it is to this rangethat most of the attention has been given. On the other hand, as iswell-known, periodic vortex shedding also occurs at higher ReynoldsProvided by IHSNot for ResaleNo reproduction or networking permitted with
18、out license from IHS-,-,-NACATN 2913 3numbers, up to 105 or more, but the free vortices which movedownstreamare quickly obliterated, by turbulent diffusion, and a turbulent wakeis established.The present interest in the vortex street is due to somequestionsarising from the study of turbulent flow be
19、hind cylinders and grids.Such studies are usually madeat Reynolds numbersfor which periodicvortex shedding from the cylinders or grid rods might occur. However,the measurementsare always taken downstreamfar enough to insure thatthe periodic velocity fluctuations are obliterated and the flow is com-p
20、letely turbulent. There are several important consequencesof thislimitation.First, the energy of the velocity fluctuations is quite low comparedwith the energy near the cylinder, and especially low comparedwith thedissipation represented by the form drag. In attaining the developeddownstreamstate th
21、ere is evidently not only a rapid redistribution ofenergy amongthe spectral componentsbut also a large dissipation.Second, the theories which describe these downstreamstages do not relatethe flow to the initial conditions except very loosely in terms ofdimensionless parameters, and it is usually nec
22、essary to determine anorigin empirically (e.g., mixing-length theory or similarity theories).On the other hand, there is evidence that somefeatures are perma-nent, so that they must be determined near the beginning of the motion.One such feature is the low-wave-numberend of the spectrum which (inthe
23、 theory of homogeneous turbulence) is invariant.Another is the random element. It has been pointed out by Dryden(references 12 and 13) that in the early stages of the decay of isotropicturbulence behind grids the bulk of the turbulent energy lies in aspectral range which is well approximated by the
24、simple functionA characteristic of certain random processes. Liepmann (refer-1 + B2n 2ence 14) has suggested that such a random process may be found in theshedding of vortices fram the grids.In short, there has been no description, other than very qualita-tive, of the downstream development of wakes
25、 which, over a wide rangeof Reynolds number, exhibit a definite periodicity at the beginning.The measurements reported here were undertaken to help bridge this gap.The main results show the downstream development of the wake, interms of energy, spectrum, and statistical properties. This develop-ment
26、 is quite different in two Reynolds number ranges, the lower oneextending from about 40 to 150 and the upper, from 300 to 104 (and prob-ably 105), with a transition range between. The lower range is theProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
27、4 NACATN 2913region of the classic vortex street, stable and regular for a long dis-tance downstream. The fluctuating energy of the flow has a discretespectrum and simply decays downstreamwithout transfer of energy toother frequencies. Irregular fluctuations are not developed. In theupper range ther
28、e is still a predominant (shedding) frequency in thevelocity fluctuations near the cylinder, and most of the emergy isconcentrated at this frequency; however, someirregularity is alreadydeveloped, and this corresponds to a continuous spectral distributionof someof the energy. Downstream,the discrete
29、 energy, at the sheddingfrequency, is quickly dissipated or transferred to other frequencies,so that by 50 diameters the wake is completely turbulent, and the energyspectrum of the velocity fluctuations approaches that of isotropieturbulence.All other features of the periodic shedding and wake pheno
30、menamaybe classified as belonging to one or the other of the two ranges. Thisviewpoint allows somesystematization in the study of wake development.In particular, it is felt that the possibilities of the vortexstreet are by no meansexhausted. A study of the interaction of periodicfluctuations with a
31、turbulent field seemsto be a fruitful approach tothe turbulence problem itself. It is planned to continue the presentwork along these lines.From a more immediately practical viewpoint an understanding of theflow close to a bluff cylinder is important in at least two problems,namely, structural vibra
32、tions in memberswhich themselves shed vorticesand structural buffeting experienced by membersplaced in the wakes ofbluff bodies. Manyof these are most appropriately treated by the statis-tical methods developed in the theories of turbulence and other randomprocesses (reference IS). These methods are
33、 easily extended to includethe mixed turbulent-periodic phenomenaassociated with problems such asthe two mentioned above.The research was conducted at GALCITunder the sponsorship and withthe financial assistance of the National Advisory Committee for Aeronautics,as part of a long-range turbulence st
34、udy directed by Dr. H. W. Liepmann.His advice and interest throughout the investigation, as well as helpfuldiscussions with Dr. Paco Lagerstrom, are gratefully acknowledged.SYMBOLSconstantsmajor and minor axis, respectively, of correlation ellipseProvided by IHSNot for ResaleNo reproduction or netwo
35、rking permitted without license from IHS-,-,-NACATN 2913 5CDCopDdddsEEI,E 2FFl(n) ,F2 (n)Fr(n)(nA)hhh*KkLLzZMkdrag coefficientform drag coefficientoutside diameter of ringcylinder dimensiondistance between free vortex layersdiameter of ring-supporting wirewake energycomponents of wake energy due to
36、periodic fluctuationsdimensionless frequency (nl-)energy spectrumenergy spectra of discrete energycontinuous energy spectrumoutput of wave analyzer at setting nAlateral spacing of vorticesinitial lateral spacing of vorticeslateral spacing between positions of umconstantintegerscalescale correspondin
37、g to Rzdownstream spacing of vorticesmoment_ of order k, of probability densityProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 NACATN 2913NknIn 2 = 2n 1qq(r)q* = q(r*)RR(n)RhRt(T)Rzrr _SS ITTatUUoU*“absolute“ moment of probability densityshedding f
38、requencyprobability distribution functionprobability densityarea under response characteristictangential velocity in vortexReynolds numberresponse characteristic of wave analyzerReynolds number based on ring diametertime correlation functionspace correlation functiondistance from vortex centerradius
39、 of vortexnumber, based on cylinder dimension (nld/Uo)StrouhalStrouhal number_ based on distance between f_ee vortexlayers (nld/Uo)time scaletime of averagingtimelocal mean velocity in x-directionmean stream velocitymean velocity at vortex centerProvided by IHSNot for ResaleNo reproduction or networ
40、king permitted without license from IHS-,-,-NACATN 2913 7U,V,WUl,U 2u rU mVx,y,zF5(n)nv00TO3components of velocity fluctuationperiodic velocity fluctuations, at frequencies nI and n2random velocity fluctuationpeak root-mean-square value of velocity fluctuationvelocity of vortex relative to the fluid
41、refe_nce axes and distance from center of cylinderflatness factor of probability distribution _4/M2 2)strength (circulation) of a vortexDirac delta functionpositive numberdistance between two points, measured in z-directiondimensionless frequency (_o n)dimensionless “time“ in life of vortex _-_dummy
42、 variablekinematic viscositya value of udensityskewness of probabilitydistribution (M3/M23/2)time intervalhalf band width of wave analyzerProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NACATN 2913GENERALCONSIDERATIONSExcept for the parameters dire
43、ctly related to the shedding frequency,the quantities measuredwere essentially those that are standard inturbulence investigations (cf. references 12 to 14). These are brieflyreviewed below with somemodifications required to study the periodicfeatures.Reference AxesThe origin of axes is taken at the
44、 center of the cylinder (fig. i);x is measureddownstreamin the direction of the free-stream velocity,z is measuredalong the axis of the cylinder, which is perpendicularto the free-stream velocity, and y is measuredin the direction per-pendicular to (x,y); that is, y = 0 is the center plane of the wa
45、ke.The free-stream velocity is Uo and the local meanvelocity in thex-direction is U. The fluctuating velocities in the x, y, and zdirections are uj v, and w, respectively. The flow is consideredto be two-dimensional; that is, meanvalues are the samein all planesz = Constant.Shedding FrequencyThe she
46、ddingI frequency is usually expressed in terms of the dimen-sionless Strouhal number S = nld/U o, where nI is the shedding fre-quency (from one side of the cylinder), d is the cylinder diameter,and Uo is the free-stream velocity. The Strouhal number S may dependon Reynolds number, geometry, free-str
47、eam turbulence level, cylinderroughness, and so forth. The principal geometrical parameter is thecylinder shape (for other than circular cylinders, d is an appropriatedimension). However, cylinder-tunnel configurations must be taken intoaccount, for example, blockage and end effects. In water-channe
48、l experi-ments surface effects mayhave an influence. Usually the geometricalconfiguration is fixed, and then S is presented as a function ofReynolds number R.Instead of Strouhal number it is sometimesconvenient to use thedimensionless parameter F = nld21v, where v is the kinematic viscosity.IThe term “shedding“ is used throughout this report, for convenience;it is not meant to imply anything about the mechanismof the formation offree vortices.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-_H NACATN 2913 9EriergyThe experiments to be des
